I got this directly from the Spinning Web site, www.spinning.com
“Pedaling backwards is risky on a fixed gear bike. If riders try to quickly stop the flywheel while pedaling backwards, the compressive forces on the knee joint can be sufficient enough to tear cartilage or the meniscus. Also, pedaling backwards may hyperextend the legs, which could damage the ACL or other soft tissue of the knee joints. Aside from being risky, a further reason not to do it is that there is no physiological advantage to it. A study in the Strength and Conditioning Journal showed that muscle contribution and metabolic cost were the same for pedaling forward and backward. Lastly, this movement puts the bike at risk as well. Pedaling backwards may eventually unscrew the pedals from the crank arm.”
I would also check with your bike manufacturer b/c it may actually do damage to your bike/cranks etc. I don’t recommend or teach it, although some students do include it as part of their cool down, seated and at a low tension. I would discourage anything else.